OFFSET
0,3
COMMENTS
Number of 0..n-1 arrays of 5 elements with zero 2nd differences. - R. H. Hardin, Nov 15 2011
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Daniel Gabric and Joe Sawada, Investigating the discrepancy property of de Bruijn sequences, University of Guelph (Canada, 2020).
János Pach and Pankaj K. Agarwal, Combinatorial Geometry, p. 220, 1995, Problem 13.10.
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).
FORMULA
G.f.: x*(1+x^4)/((1-x)^2*(1-x^4)).
a(n) = 2*a(n-1) -a(n-2) +a(n-4) -2*a(n-5) +a(n-6). - R. H. Hardin, Nov 15 2011
a(n) = (-2*(1+(-1)^n)*(-1)^floor(n/2) + 2*n^2 + 5 - (-1)^n)/8. - Tani Akinari, Jul 24 2013
E.g.f.: ((2 + x + x^2)*cosh(x) + (3 + x + x^2)*sinh(x) - 2*cos(x))/4. - Stefano Spezia, May 26 2021
Sum_{n>=1} 1/a(n) = Pi^2/24 + tanh(Pi/2)*Pi/4 + tanh(sqrt(3)*Pi/2)*Pi/sqrt(3). - Amiram Eldar, Aug 25 2022
a(n) = 2*floor((n^2 + 4)/8) + (n mod 2). - Ridouane Oudra, Sep 08 2023
MAPLE
f := n->n^2/4+3*n/2+g(n);
g := n->if n mod 2 = 0 then 3 elif n mod 4 = 1 then 9/4 else 13/4; fi;
seq(f(n), n=-3..50);
MATHEMATICA
CoefficientList[Series[x*(1+x^4)/((1-x)^2*(1-x^4)), {x, 0, 60}], x] (* G. C. Greubel, Sep 12 2019 *)
PROG
(PARI) concat([0], Vec(x*(1+x^4)/((1-x)^2*(1-x^4))+O(x^60))) \\ Charles R Greathouse IV, Sep 26 2012, modified by G. C. Greubel, Sep 12 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0] cat Coefficients(R!( x*(1+x^4)/((1-x)^2*(1-x^4)) )); // G. C. Greubel, Sep 12 2019
(Sage)
def A008811_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P(x*(1+x^4)/((1-x)^2*(1-x^4))).list()
A008811_list(60) # G. C. Greubel, Sep 12 2019
(GAP) a:=[0, 1, 2, 3, 4, 7];; for n in [7..60] do a[n]:=2*a[n-1]-a[n-2] +a[n-4]-2*a[n-5]+a[n-6]; od; a; # G. C. Greubel, Sep 12 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved